Combine potatoes and eggs in the shell with enough water to cover (salted is optional) in a saucepan. Heat to a gentle boil over medium-high heat. Cook 8 minutes. Remove eggs and rinse under cold water to stop cooking. Set aside; once cooled, peel off shells and quarter.

Add green beans to saucepan with potatoes and continue cooking until vegetables are fork-tender, about 8 minutes more. Drain and cool 5 minutes.

Drain oil from tuna, reserving 3 Tbsp. Whisk vinegar and mustard into oil until well blended. Arrange all salad ingredients on a platter and drizzle with dressing.

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